J. Phys. Chem. C 2008, 112, 6667-6676
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Theoretical Simulation of AlN Nanocrystals Aurora Costales,* M. A. Blanco, E. Francisco, C. J. F. Solano, and A. Martı´n Penda´ s Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Quı´mica, UniVersidad de OViedo, 33006-OViedo, Spain ReceiVed: April 26, 2007; In Final Form: February 14, 2008
The periodic cluster model, a theoretical scheme recently developed in our laboratory, has been used, together with an atomistic pair potential energy model, to simulate AlN nanocrystals. The main advantages of this approach are a greatly reduced dimensionality and the ability to incorporate external pressure effects at the cost of neglecting geometric surface relaxation. The results suggest that small-size nanocrystals display graphiticlike layers, nonbuckled, as opposed to the buckled bulk wurtzite structure; buckling will dominate after a given, quite large, threshold. The surface tension, c/a ratio, and equation of state are examined with respect to variations on both pressure and nanocrystal size. It is found that, although the bulk modulus of the wurtzitelike phase is smaller (although decreasing with size) in nanocrystals than in the bulk, the nonbuckled phase displays a larger modulus, in agreement with the experiments. The transition pressure into the rocksalt-like phase increases with system size, with the dependence being linear against 1/N 1/3. This is rationalized through a qualitative small-systems thermodynamical model.
I. Introduction There is a growing interest on nanocrystals, due to their novel electronic and mechanical properties.1 The tailoring of materials with a wide range of properties can be achieved by tuning the size of the nanocrystals of a given compound. Thus, investigating the size dependence of their properties has become a basic science task with important applied consequences. It is particularly interesting to compare the nanocrystal behavior with the bulk crystal one, since they may even present different crystalline structures. The pressure behavior, related to hardness and mechanical properties but also to structural transformations, is one of the key variables to explore. Despite a large number of experimental works in the area,1-8 theoretical nanocrystal works are more scarce9-16 and these include limited nanocrystal sizes (up to 5000 nanocrystal atoms). The reason for this is twofold. On the one hand, although nanocrystals display some degree of periodicity, their finiteness is their key characteristic, and so periodic crystal simulation techniques do not apply. On the other hand, although usual molecular or cluster techniques do apply in principle, the size of nanocrystals lies in the 1-20 nm range; this means thousands or millions of atoms, a great challenge to most methods. There are two problems associated with these large sizes. (i) The most accurate electronic structure methods cannot cope with such system sizes, since the computational costs of energy evaluation scale with a high power of system size and with large prefactors; hence, accuracy has to be sacrificed to reduce the computational cost. (ii) For any given method, the search for a minimum energy structure has a computational cost that increases exponentially with the number of free variables (see, for example, the excellent series of papers by Hamada et al.17-19 on clusters of gradually increasing size or our study on ref 20). It has to be remarked that dynamical simulations, although useful in some respects, only follow the forces from a predefined configuration; it is * To whom correspondence should be addressed. E-mail: yoyi@ carbono.quimica.uniovi.es.
the selection of the initial configuration that lies behind the problem of global optimizations. Hence, new modelization techniques that overcome these limitations for the simulation of nanocrystals are needed. The first goal of this article is to present a recently developed21 theoretical scheme, the periodic cluster model, designed to address the above limitations (see section II). It does so by using the periodicity displayed by the nanocrystals to reduce the dimensionality of the configuration space; thus, it overcomes limitation (ii) above. However, it maintains the finiteness of the system, and although it can be applied also to high-accuracy methods, in practice, one is forced to trade some accuracy in exchange for reaching the large-size nanoscale limit, so limitation (i) still applies. In addition to its computational feasibility, there is an important feature of the periodic cluster model that makes it specially appealing: it allows the definition of a volume, and hence a pressure, that may be applied in the microscopic simulation of these finite systems (see ref 22, where different methods are compared). This is not easily achieved in other approaches, which often utilize a noble-gas pressure medium much larger than the nanocrystal to be simulated (350 000 atoms for a nanocrystal of 4500 atoms,13,23 150 000 for 705,10 30 000 for 4000,11,14 and 3 000 000 for 40 00012). The second goal of the article is to employ the abovementioned scheme in the study of a controversial problem in semiconductor nanocrystals. There were several studies3,4,6 in which the pressure behavior of the nanocrystals had a common trend: the bulk modulus of the material was enhanced as size decreased, while the phase transition pressure increased. However, the nanocrystals of other substances presented the opposite phase transition pressure trend,5 notably AlN.8 This is a technologically important material (see, e.g., refs 24-26) in which our group has previous experience in the simulation of the electronic structure of its small- and medium-sized nanoclusters,27-29 the bonding at both the molecular and crystalline levels,30 and the global optimization of large-size nanoclusters.20
10.1021/jp073228i CCC: $40.75 © 2008 American Chemical Society Published on Web 04/04/2008
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The rest of the article is organized as follows. Section II presents our simulation methodology, including the periodic cluster model and its application in the simulation of the pressure behavior. Section III presents the results of our simulations on AlN nanocrystals, including its behavior upon size, pressure, and phase variation. Finally, section IV presents our main conclusions and gives some prospects for future works. II. Methodology The simulation methodology employed in this work comprises three main pieces: First, the well-known pair potential scheme to evaluate the energy of a given configuration. Second, a novel periodic cluster approach to the selection of configurations in nanocrystals, in which the energetics of the surface termination is included without its geometric complexities. Third, the inclusion of the external pressure, a thermodynamic variable, within the microscopic simulation in this periodic model. A. Pair Potential Scheme. The basic tenet of this scheme is that the (binding) energy surface of a system can be computed as a sum of pair potential interatomic interactions,
E)
1 2
Vij(rij) ∑i ∑ j*i
(1)
The pair potential, Vij(rij), is assumed to be central in most cases, although polarization effects can be incorporated in various ways, being perhaps the most popular the shell model originally proposed by Dick and Overhauser (ref 31, but see also refs 32 and 33). In this work, we will mainly use the central potentials derived in our study of the evolution with size of the properties of AlN clusters toward their bulk values (ref 20). These potentials were ab initio derived (thus being widely transferable) from in crystal atomic descriptions (thus being appropriate for nanocrystal simulations in particular). Their functional form is ionic-like, with a Coulomb potential derived from partial (not nominal) atomic charges and a short-range repulsive term, so that it can be easily evaluated in very large systems. The use of this simple model can be justified as follows. First, large nanocrystals cannot be simulated in more than a few configurations (if any) using sophisticated ab initio or semiempirical quantum mechanical electronic structure methods, thus precluding their use in the simulation of a phase transition. Hence, simplified schemes are a key in the understanding of the nanoscale, and the atomistic simulations with pair potentials are the next more accurate alternative. Second, our previous simulations of AlN nanoclusters20,29 have shown that, although much simpler, these potentials display indeed many of the same features of quantum-mechanical energy landscapes, including the relative ordering of different isomers in medium-size clusters (up to 32 atoms). We are thus confident that this scheme can be able to predict structural and equilibrium properties of AlN nanocrystals. In addition, based on similar studies,34-36 we believe that the pair potential transferability is good enough for the study of pressure-induced phase transitions. B. Periodic Cluster Model. Even under the assumption of pairwise additivity of central potentials, the problem of exploring the potential energy surface to select lowest-energy configurations is still an enormous task even for medium-sized nanoparticles.20 Although the energy evaluation for a given configuration can still be achievable, the number of degrees of freedom, roughly 3Nt (Nt being the total number of atoms), becomes so large that either (i) the convergence of energy minimizations is very slow if gradient-only techniques are used, or (ii) the size of the problem becomes easily intractable if quadratic-
convergence Hessian matrix optimization techniques are used. Thus, even a single optimization run is a troublesome task, and global optimizations are out of reach. In the simulation of solids and liquids, these dimensionality problems are solved by using periodic boundary conditions and an infinite size assumption: in the solid, the crystalline structure displays indeed a unit cell that is replicated in the three Cartesian dimensions, while in the liquid the simulation cell is taken as large as possible so that this replication does not affect the average properties. However, the problem is different in nanosystems, where the size is necessarily finite. Even in nanocrystals, where the experimental structure displays some degree of periodicity, it is the frontier effects that distinguish them from macroscopic crystals, and hence, an infinite replication cannot be used. In order to avoid these problems, our nanocrystal simulations employ the periodic cluster model, a recently developed21 theoretical scheme implemented in the computer code cluster.37 In this model, the configurations available for the energy minimization are restricted to those described by a periodic lattice, hence reducing the number of degrees of freedom to those in the description of the unit cell. On the other hand, being a nanocrystal, the replication of this unit cell is not infinite but finite in the three dimensions. This introduces the frontier effects within the energy of a given configuration. However, no geometric frontier effects are considered here: the atoms in the outermost cell are still restricted to follow the rigid structure of the periodic lattice. This, in fact, corresponds to minimizing the energy of unit cells subjected to a kind of average Madelung potential obtained from considering all their possible different spatial environments. Although mixed models can be devised, in which the frontier cells are allowed to lose part or all of their periodicity (perhaps defining face, edge, and vertex regions, or maybe crystalline domains38), this has not been done in the present initial study. Thus, the nanocrystal is assumed to have a unit cell with N atoms, which is replicated in space to fill the shape of an actual nanocrystal. In this work, we will restrict to parallelepiped shapes, in which the unit cell is replicated an integer number of times on each direction, so that there are Ncells ) Na × Nb × Nc cells in total. Hence, the position of atom i (i ) 1,..., N) in cell n (n ) 1,..., Ncells) will be
xi,n ) xi + ln ) (xi + hn)a + (yi + kn)b + (zi + ln)c (2) where a, b, and c are the lattice vectors defining the unit cell, xi ) (xi, yi, zi) is the fractional position of atom class i within the unit cell (all of its components lie in the [0,1) range), and ln ) (hn, kn, ln) is a vector of integers describing the position of the n cell (with components in the [0, Na - 1], [0, Nb - 1], and [0, Nc - 1] ranges, respectively). In this way, the Nt ) N × Ncells atoms have positions that depend only on the 6 independent lattice parameters, a, b, c, R, β, and γ, that define the lattice vectors,and on the 3N coordinates of the atoms in a given unit cell, at most 3N + 6 coordinates with N , Nt. Further restrictions in these variables may be adopted, perhaps to maintain the symmetry features displayed by the bulk crystal structure in the simulated nanocrystal. In this way,
E)
Vij(|xi0 - xj0|) ∑ ∑ ∑Vij(|xin - xjm|) + Ncells∑i ∑ m ni
(3)
a real-space O(N2t ) summation. No gaining is seen here, but, since E depends at most on 3N + 6 variables, gradient and
Buckled, Nonbuckled A1N Nanocrystal Structures hessians are just O (N 2t × N ) ) O (N 2cells × N 3) and O (N 2t × N 2) ) O (N 2cells × N 4) tasks, respectively, still feasible even for large Nt (or Ncells). Since, according to the experimental evidence, the nanocrystals usually display phases with geometries fairly similar to those of the bulk crystals (but see, e.g., ref 39 for counterexamples), it seems a natural choice to use the bulk crystal unit cell in the definition of the periodic cluster model. However, one must remember that (i) other cells can be devised that still reproduce the bulk crystal by periodic repetition and (ii) the choice of unit cell will determine the shape in the frontier of the parallelepiped nanocrystal. Thus, other choices may be convenient, as we shall show in the following. Since a nanocrystal with Na × Nb × Nc cells will contain different numbers of atoms for different structures or even different choices of unit cells for the same structure, it is convenient to define N, the number of units formula in the nanocrystal, as N ) ZNcells, with Z being the number of units formula in the N atom unit cell used. Thus, n ) N/NA is the mole number of the system (only useful in the macroscopic limit), and we can define an energy per unit formula, E h ) E/N, which will converge to the (binding) energy of the bulk crystal per unit formula in the h will depend on the choice of unit (Na, Nb, Nc) f ∞ limit. E cell, C, and size of the system (number of atoms, implicitly through N in C and explicitly through N ) (Na, Nb, Nc)), having the abovementioned 3N + 6 variables (at most) as configuration parameters, which we will group in a configuration vector s, so that E h C(s; N). This will allow one to exhaustively explore the potential energy surface, since the number of parameters increases only with N, and not with Ncells (although the computational cost of a configuration energy evaluation will still increase with Nt ) N × Ncells as N 2t ). C. Simulation of Hydrostatic Stress and Phase Transition. The connection between microscopic simulation and macroscopic measurements has to be dealt with carefully when small systems are considered.40 In particular, since extensive properties are no longer proportional to the number of molecules, the chemical potential, µ ) (∂G/∂n)p,T, is no longer equivalent to the molar Gibbs function, Gm ) G/n, with n being the mole number, or its microscopic analogue, the Gibbs function per unit formula G h ) G/N (µˆ in the notation of ref 40). To compare with our simulations, we have considered that the experimental nanocrystals, once grown, constitute a polydisperse mixture, with different values of N for different crystals, but that cannot exchange molecules among each other. Also, we will consider a constant temperature (T) and pressure (p) environment. This (N, p, T) environment is, we believe, a faithful representation of the high-pressure experimental conditions. Furthermore, in order to simplify the theoretical simulation, we will consider the static limit, with T ) 0 and neglecting zero-point vibrations. Thus, differences in vibrational entropies among structures are assumed negligible when considering the relative stability of competing polymorphs. Under the above assumptions, G h )E h + pV h , where V h ) V/N is the volume occupied by the nanocrystal per unit formula and p is the external parameter. While it is difficult to define a consistent molecular volume in general, the periodic cluster model provides a quite simple answer: the unit cell volume, divided by Z, gives an approximate value that correctly converges to the bulk limit when (Na, Nb, Nc) f ∞. Although the boundary of the nanocrystal is ill-defined due to the fuzziness of the atomic volume concept in isolated systems, the use of cell volumes, with atoms presumably compacted at the volumes appropriate for the given pressure, seems quite sensible. Thus,
J. Phys. Chem. C, Vol. 112, No. 17, 2008 6669 we can compute G h for any configuration of any choice of unit cell and replication numbers at any given pressure,
G h C(s; N, p) ) E h C(s; N) + p V h C(s)
(4)
even outside of equilibrium. The equilibrium condition for a (N, p, T) environment, that G h must be a minimum with respect to internal variables, leads to
seq(N, p)| G h C(s; N, p) minimum
(5)
h C(seq; N, p). We can restrict further which defines G h C(N, p) ) G this equation: since there can be different phases, different unit cell choices C for a given phase, and possibly different N leading to the same N, or combinations of these, the real equilibrium value will be achieved by the lowest G h shape and phase, so that the final variables are just N and p,
Neq(N, p)
and
Ceq(N, p)| G h C(N, p) minimum
(6)
and G h (N, p) ) G h Ceq(N, p)(Neq(N, p), p). Partial optimizations obtaining the minimum energy shape for a given phase, N, and p are possible, defining the G h value for the phase. It is also worthwhile to compute the surface tension, γ, of the nanocrystal against vacuum; this can be defined by the macroscopic equation (static limit)
G ) U + pV + γA
(7)
where U ) NU h. U h , the macroscopic analogue of E h , can be computed through a perfect crystal simulation with the same unit cell geometry. A is the external surface area of the nanocrystal and will be computed as the sum of the areas defined by the parallelograms on the borders of the nanocrystal, in accordance with our previous volume definition. Thus,
γ ) (E h -U h )/Ah
(8)
is the excess surface energy per unit area for a given nanocrystal configuration, where Ah ) A/N. The procedure described by eq 5 has been implemented in our cluster program,37 so that properties for given C, N, and p can be obtained. We will also compute G h values for the phases by performing a limited shape minimization: only a few choices of unit cells will be taken, and a given algorithm for selecting N values employed, selecting the lowest-G h one for each N (or range of N values) as the minimum for the phase. This procedure cannot be justified completely, but it is expected that shape effects (nevertheless only partially represented in this fixedgeometry boundary model) will have a small impact in the thermodynamics phase transition (although see refs 41 and 42 for kinetic effects), with the above procedure being appropriate for a first estimate. We will test this assumption by comparing the results for different shapes. The main results here, the V h φ(N, p) equations of state and the G h φ(N, p) free enthalpy functions for the different phases φ, can then be used to obtain the transition properties from phase 1 to phase 2 by applying the h 2(N, ptr), equilibrium condition. This condition is G h 1(N, ptr) ) G implicitly defining ptr(N); from this function, other transition properties such as ∆V h tr(N) ) V h 2(N, ptr) - V h 1(N, ptr) can be obtained. Finally, it is important to notice the following properties for finite systems. On the one hand, finite systems at finite temperature do not display sharp phase transitions, and the experiments do indeed display fuzzy transitions. However, appropriate transition magnitudes can be defined in this case by using a two-state approximation.40 On the other hand, finite
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systems in the static limit do again display sharp boundaries analogous to those in the macroscopic limit.40 Nevertheless, the transition properties defined above should approximate the experiments correctly provided temperature effects are small. III. Results and Discussion The previous techniques have been applied to the simulation of nanocrystals of aluminum nitride, AlN. We will present our results as follows. First, we will describe the nanocrystals simulated, with different unit cells leading to varying phases and shapes; in all cases, we have chosen a particular growth pattern, so as to capture the main trends in size effects without being exhaustive. We will then show how including pressure affects the nanocrystals, describing the equations of state of the main crystal phases. Finally, we will address the phase transition between the B4 and B1 nanophases, which should be the finitesize analogue of the well-known43-46 bulk crystal B4 h B1 phase transition induced by pressure. A. Computational Growth of Nanocrystals. Our aim here is to sample the optimum structures for the different phases and how these evolve with size. As presented in section II, we do so by selecting a unit cell and periodicity repetition numbers Na, Nb, Nc, which in turn fix the number of unit formulas (or molecules) in the nanocrystal, together with its shape. For each of these choices, we optimize the pair potential energy with respect to the structural parameters, with these being the unit cell lengths and angles, together with the coordinates of the atoms in the cell, although some of these may be fixed in order to better reproduce the structures and simplify the calculation. This will give the zero-pressure static minima within this restricted parameter space. A sample of each of the different nanocrystal classes employed in our simulations, corresponding to different cell choices, is presented in several projections within Figure 1. First, a natural choice of unit cell for B4-like nanocrystals is the bulk B4 unit cell. This is a hexagonal-system cell, with γ ) 120° and two molecules per cell (Z ) 2), which we have denoted as B4h (see Figure 1a and b or c depending on the optimization of z; see below). In our simulation, we have restricted the cell to remain hexagonal, with fixed angles and a ) b, while varying a and c. In the B4h cell, we have also defined another internal coordinate: both N atoms are allowed to move along the z axis, although keeping their mutual distance. These are the same free parameters that the bulk B4 structure displays. The B4hf cell uses the same definition, but fixing z to its value in the B4b computed B4 bulk structure. Thus, Figure 1b corresponds to the optimized-z B4h, while Figure 1c corresponds to the fixed-z B4hf. As a second unit cell choice, we have selected an orthorhombic unit cell, B4o, which seems to be the periodic repetition unit in several other B4-like nanostructures, such as ZnO nanobelts and nanorings.47 It corresponds to a simple transformation of the hexagonal one, with its cell axes being 2a + b, b, and c, this time with Z ) 4 (see Figure 1d, b or c, and e or f depending again on the optimization of z). To keep the structure simple, we take as variables a common z displacement for the four N atoms and the three cell lengths; however, we have found that the optimized ratio of the axes lying on the (001) plane is always x3, the one recovering the B4-like hexagonal rings, to an accuracy better than 1%. As with the previous cell, we have also defined a fixed-z cell denoted as B4of. Thus, Figure 1b and e correspond to optimized-z B4o, while Figure 1c and e correspond to the fixed-z B4o. Regarding the B1-like nanocrystals, we have again two choices, with Z ) 4 and Z ) 2. The first one is the conventional cubic unit cell,
Figure 1. Two-dimensional projections of nanocrystals employed in the simulations, according to different cell choices as described in the text. Empty circles represent atoms behind the projection plane. The N ) (Na, Nb, Nc) values of the actual nanocrystals represented are (5, 5, 3) for B4h and B4hf, (3, 5, 3) for B4o and B4of, (4, 4, 4) for B1c, and (5, 5, 4) for B1t. The different views correspond to the following nanocrystal faces: (a) (001) of B4h and B4hf; (b) (100) ) (010) of B4h and (100) of B4o; (c) (100) ) (010) of B4hf and (100) of B4of; (d) (001) of B4o and B4of; (e) (010) of B4o, corresponding to a (210) face in the Bk structure; (f) (010) of B4of, corresponding to a (210) face in the B4 structure; (g) (100) ) (010) ) (001) of B1c; (h) (001) of B1t; and (i) (100) ) (010) of B1t, corresponding to a (110) face in the B1 structure. The projections of the unit cells are also included. The B4h and B4o nanocrystals are represented with the optimum, Bklike, z ) 0 value (see text).
B1c, in which we only allow for the variation of the a ) b ) c parameter (see Figure 1g). The second one, a tetragonal cell with (a - b)/2, (a + b)/2, and c axes labeled B1t (see Figure 1h and i), was chosen to be analogous to the B4h cell in the B1 structure. The internal parameters were fixed, while the cell was restricted to be tetragonal; however, the optimized axis ratio was always x2 to a 2% accuracy, thus maintaining the B1-like structure. To summarize, B4h and B4o stand for B4-like nanocrystals with hexagonal and orthorhombic cells, respectively, optimizing z; B4hf and B4of are those that fix z to the B4b value; B1c and B1t are the cubic and tetragonal cell B1-like nanocrystals. For the sake of completeness, we will also use the notations B4b, B1b, and Bkb for computed bulk phases, B4e for the experimental B4 bulk results, and B4n for experimental B4-like nanocrystals. For each of these unit cells, we have selected nanocrystals with repetition numbers according to a single size parameter, k. For the Z ) 4 structures, B4o, B4of, and B1c, we have chosen (Na, Nb, Nc) sizes given as (k, k, k) for increasing values of k. In the case of the Z ) 2 structures, B4h, B4hf, and B1t, we have selected a (k, k, l) scheme, where l has been taken as the nearest integer to k/x2 so that the shape of the crystal is as compact as possible. This would lead to the lowest energy if all faces were equal in energy per unit area. This is not always the case, but we will not extend our search for the lowest energy in that direction. Nevertheless, the experimental nanocrystal shapes are not known, and furthermore, they are frequently under kinetic control. Hence, the optimum shape may not be the best one to compare to the experiments.
Buckled, Nonbuckled A1N Nanocrystal Structures
Figure 2. Binding energy per molecule (E h , in Eh per molecule) vs number of AlN molecules (N) in the nanocrystal for different cell choices. Solid lines correspond to B4 nanocrystals: squares for B4o and triangles for B4h. Dashed lines correspond to B1 nanocrystals: + symbols for B1t and × sumbols for B1c. Horizontal lines are the B1 and B4 bulk limits. The inset shows the same nanocrystals, together with the fixed-z B4hf and B4of nanocrystals, which are represented by empty triangles and squares, respectively.
Figure 2 summarizes the energy results of the optimizations for each of these series of nanocrystals at 0 GPa. The energy per unit formula, E h , is represented, and so its value for all sizes and phases should be on the same order of magnitude. We can see that B4h and B4o converge to minima very similar in energy when size increases, with B4h being slightly higher in energy (less than 1 kcal/mol). On the other hand, the minima for the B1c structure also stabilize in the size range shown to a value that differs by about 2 kcal/mol from those of the B4-like nanocrystals, while the B1t limit seems to be some 3 kcal/mol above the B1c one. So, it seems that the different B4 faces do indeed have a similar surface energy, somewhat smaller for the (210) face than for the (100) one, but this is not the case for the B1 faces: the (110) face of the B1t nanocrystals seems to be higher in energy than the (100) faces of the B1c ones (see Figure 1). This can also be deduced from the steps shown in the B1t structure: for some values of k (e.g., 11 and 12), l is the same (8 in the above case), and, although the surface per molecule continues to decrease, the energy lowering per unit surface is much smaller (see the γ discussion below). There is an important characteristic in the B4-like nanocrystal results. All of the energy optimizations lead to a z ) 0 value; that is, Al and N atoms lie in the same plane instead of having a buckled structure (chairlike hexagons) as in the B4 crystal. All of these optimizations started with the z value of the crystal structure, which casts some doubt over its minimum character for these nanostructures using this potential model. Therefore, we have examined the B4 nanostructures fixing the z value to the buckling in the crystal structure, the B4hf and B4of curves. They are higher than B4h/B4o curves by more than 36 kcal/ mol, as seen in the inset of Figure 2 (although both still lie within 1 kcal/mol of each other). This anomalous behavior is due to their polar surfaces (see Figure 1c and f for B4hf/B4of, compared with b and e for B4h/B4o) being highly energetic, which also lead to several other unphysical trends for increasing sizes (see below). On the other hand, in the infinite-crystal bulk limit with no surface effects, the computed B4 phase (B4b) displays a z ) 0.1098 value, close to the experimental B4 phase (B4e) value, z ) 0.1128. However, the z ) 0 bulk structure, which will be labeled as Bkb, lies 1 kcal/mol below the B4b structure using the present potential model; although they are almost degenerate, this is a drawback that we have to address. To do this, and to test whether the nonbuckled layers are the genuine structure of the nanocrystals, we have considered four possible limitations in our simulation: (i) the lack of polarization effects, (ii) the lack of geometric relaxation of the surface layers
J. Phys. Chem. C, Vol. 112, No. 17, 2008 6671 in our periodic model, (iii) the lack of electronic structure relaxation, and (iv) the existence of important surface defects in the (001) surfaces, (v) a limitation which is not examined here (and is excluded by the construction of the calculation) is the appearance of nonperiodic geometries or separate crystalline domains. Regarding the effect of polarization, we have used two different shell-model interaction potential sets derived for AlN.48 In both cases, the B4 minimum is lower than the Bk one in the bulk structures, clearly showing that the lack of polarization is behind the above inverted bulk preference. However, all the nanostructures studied still lead to z ) 0 minima even though they start as B4-like structures. With respect to the lack of relaxation in the surface layers due to the imposed periodicity, preliminary results on AlN nanobelts using the same potential model, but in which periodicity was not imposed, displayed also a nonbuckled structure.49 The nonbuckled structure also appears at high pressure during the course of molecular dynamic simulations of the B4 h B1 phase transition of small CdSe nanocrystals,13 using empirical potentials (ionic plus LennardJones form) fitted to reproduce phase parameters and relative energy orderings of the phases involved.50 It has also been found as an intermediate when relaxing the as-cleaved nanocrystals examined in ref 15. Regarding electronic structure relaxation effects, in our density functional theory (DFT) studies of small and medium-sized nanoclusters of AlN, the rings did not display buckling, even when piled on top of each other.27-29 Also, a recent DFT study51 of nonbuckled graphitic-like versus buckled wurtzite-like films in AlN and several related compounds has shown that supercell slab optimizations lead to nonbuckled Bklike results for slabs with fewer than 24 hexagonal layers (on the order of 5 nm). The analysis of the charge distribution and electronic bands showed that, for the dipolar buckled structure to be stabilized, a significant charge transfer from the Al atoms in the bottom layer to the Al atoms on the top layer has to occur, mitigating the dipole but leading to a metalization of the slab. We have not tested whether the addition of defects in the surface leads to stabilization of the B4-like nanocrystals; however, defects do not seem to be present to a large extent in the experiments, where the transmission electron microscopy images display rather clean surfaces.52,53 The above facts point out that there is a competition between the buckled and nonbuckled structures for bulk crystals, 2D slabs, nanoclusters, and nanobelts; it seems plausible that the competition exists in clean-surface nanocrystals, too. As further supporting evidence, the nonbuckled Bk structure is also present in the same-group isovalent compound BN, which displays a B4 high-pressure phase in addition to the B1 phase. We conclude that, although the 1 kcal/mol stability of the nonbuckled structure is certainly an artifact of the potential model for the bulk results, nonbuckled structures seem to be the genuine lower energy result, at least for a certain range of nanocrystals. However, see ref 38 for the possibility of having a segregated-domains cancellation of the dipole in nanorods. To justify this result, it appears that, for low-dimensional systems, the polar surface and the 3-fold coordination of the last buckled layer are severe energetic handicaps, which force the structure toward planarity. In this way, 4-fold coordination is achieved for the surface, and 5-fold (3 + 2) coordination is achieved for the interior. Thus, we believe that these nonbuckled structures are indeed dominant, at least for sizes smaller than a given threshold. Let us recall that buckling of AlN slabs happened for sizes having more than 24 layers, and it was driven by a charge transfer between the end Al layers, stabilized
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Figure 3. Binding energy per molecule (E h , in Eh per molecule) vs the inverse of the cubic root of the number of AlN molecules (1/N1/3) in the nanocrystal for different cell choices. Symbols as in Figure 2. The inset displays the large-N region enlarged, together with the bulk limits for phases B1, B4, and Bk.
through metalization.51 However, this result corresponds to 2D infinite slabs, whereas nanocrystals are finite in all three dimensions; hence, metalization will be harder to achieve, and the nonbuckled nanocrystals will persist to a larger threshold. Since this kind of nanocrystals will have the Bkb phase as the bulk limit, we will include this in our further discussion together with the B4b buckled bulk limit. The size evolution displayed in Figure 2 by all structures seems to have a common underlying law. In fact, a common small-system model energy is40
G(N, p, T) ) N µ∞(p, T) + a(p, T) N2/3
(9)
where µ∞(p, T) is the bulk limit chemical potential and the last term is a surface tension/surface energy contribution. It is equivalent to the surface term in eq 7, since, at constant molar h and G h dependence volume, A ∝ V 2/3 ∝ N 2/3. This leads to an E which is linear in 1/N1/3; Figure 3shows a plot of this relation. First, note that all of the nanocrystals studied seem to fulfill this linear relationship for large enough N. Second, notice in the inset how, being almost linear and having limNf∞ N-1/3 ) 0, it is easier to extrapolate the bulk limit in this representation. In particular, both B1 nanocrystals tend to the B1b limit. However, B4o and B4h seem to tend to the Bk bulk instead of the B4 bulk. This is not surprising, since these nanocrystals display z ) 0 structures in all our calculations; the onset of B4-like behavior will show here as a clear change in slope, which is absent up to our largest calculation (13 500 molecules for B4o, 27 702 molecules for B4h). On the other hand, the fixed-z B4of and B4hf nanocrystals go to a limit much higher than the rest. This is because the energy of the polar (001) face is simply proportional to the size of this surface, increasing with size when no dipole compensation is present. Another way to present the surface effects is to compute the surface excess energy per unit area or surface tension, γ (see eq 8). This property is depicted in Figure 4 against N-1/3. Note that, since the main scaling of E h-U h is N-1/3, and that of A h -1/3 is N too, γ should not vary too much with N. The changes are certainly small, but even these seem to be (aside from the jumps to be explained in the next paragraph) linear with N-1/3 again in the large sizes regime, a fact for which we have found no explanation thus far. Notice that, at small sizes (right side of the figure), it has low (even negative) values, since the surface of a small cluster is ill-defined. However, it displays a convergent trend while increasing the nanocrystal size, in which it decreases while the size increases toward the bulk limit. Notice that this is not true of the B4hf and B4of curves in the inset that diverge: this is again because of the uncompensated polar
Figure 4. Surface tension (γ, in mN/m) vs 1/N1/3 for the B4h, B4o, B1t, and B1c nanocrystals. The inset displays them again in a scale that suits B4hf and B4of, too.
surface of these fixed-structure nanocrystals. This divergence is known as Tasker’s rule.54 Overall, the two Z ) 4 series of nanocrystals, B1c and B4o, present a lower γ value than the corresponding Z ) 2 ones, B4h and B1t, which gives an idea of the relative stability of the different surfaces involved. In fact, further information can be extracted from the jumps in the γ values for the Z ) 2 series. Since the growth algorithm, aiming at more compact nanocrystals (i.e., more similar to the minimum surface/volume solid, the sphere), uses the nearest-integer step function, the (k, k, l) series has l values of 1, 1, 2, 3, 4, 4, 5, 6, 6, 7, ... for increasing k values. Thus, for example, the k ) 8, 9 nanocrystals have Na and Nb values increasing by one but the same Nc ) l ) 6 value; these are the jumps exhibited by the B1t and B4h lines, when changing from a +1 increase in the three axes to an increase in just two axes. In the E h graphs, the jumps lead to a second value (for k ) 9) slightly above the previous trend; however, upon division by A h , the γ graph shows the opposite trend. Thus, the (001) faces (Figure 1a and g) have a lower contribution to γ than the (100) (Figure 1b, B4h) and (110) (Figure 1i, B1t) faces. That is, the x and y charge alternating structure of the (100) ) (001) face in B1 has a lower energy per unit area than that of the (110) face, where charge alternation only happens along the z direction. A similar thing happens with the (001) face of the Bk-like B4h nanocrystals, planar and with alternating signs, as compared with the (100) face of the same structure, alternating charges only along the c axis. After all of the above results, we can conclude that B4o and B1c nanocrystals are our best candidates to represent B4-like (Bk-like, in fact) and B1-like nanocrystals. Although we have not conducted an extensive search, (i) the (100) face is a lowenergy one, being displayed by many B1 crystals, as the cubelike rock-salt mineral, and (ii) the E h and γ differences are relatively small for the B4 crystals. Thus, despite B4o and B1c being among the most simple cell choices, they should produce a faithful representation of B4- and B1-like nanocrystals in what follows. Both nanocrystal series have Z ) 4 unit cell descriptions, and the set of sizes has been selected so that the same numbers of AlN units are sampled in both cases (this is why a most-compact algorithm has not been used for the B4o series), and thus, they are directly comparable for each size. Although this is not strictly necessary, it gives better numerical results. B. Equation of State of B4 and B1 Nanophases. Applying external pressure to the nanocrystals, in principle, in a hydrostatic manner, will lead to changes in their properties according to their corresponding equations of state. Let us start by considering again the surface tension, γ. In Figure 5, we can see that the evolution is mostly linear with pressure in all cases. The size change, however, follows the pattern seen in Figure 4: the smallest size gives a small γ value, quite separated from
Buckled, Nonbuckled A1N Nanocrystal Structures
Figure 5. Evolution of the surface tension γ (in mN/m) vs pressure (in GPa) for B4o (solid lines) and B1c (dotted lines) nanocrystals of selected sizes.
Figure 6. Evolution of the c/a ratio with p (in GPa) for B4o (solid symbols) and B4of (empty symbols) nanocrystals of selected sizes. The computed bulk behavior for the B4 and Bk phases (solid lines) is also plotted, together with the p ) 0 experimental B4 value.
the trend of medium to large sizes, which are larger than it and decrease with size. Not surprisingly, the slopes against pressure are larger (in absolute value) for the smaller crystals, so that many γ(p) lines cross each other. However, these slopes tend to very similar values for the larger nanocrystals, having almost coincident γ(p) lines. In the case of the B4-like nanocrystals, there is an important structural parameter that can vary with pressure, too (the c/a ratio). Its pressure behavior is plotted in Figure 6 for the B4o and B4of nanocrystals. Notice how the smallest size has the largest ratio on each set, with the N ) 32 size being the lowest, and from there the ratio increases with size. The computed ratios increase slightly with pressure, with all of them having a similar slope. Unsurprisingly, the B4o ratios seem to lead to the Bkb behavior rather than to the B4b one, whereas the B4of nanocrystals, with internal coordinates fixed to those of the B4b bulk, have values similar to those of B4b. However, the pressure behavior is quite different: the B4o nanocrystals and Bkb solid share a common increasing c/a trend; the B4of nanocrystals also increase their c/a ratio, while the crystal shows a decreasing trend. Experimentally, the bulk solid c/a also decreases,43 although it displays a concave curvature instead of the convex one here. This may be related to our B4 bulk instability against the Bk phase (too early in our potential model), but this needs not be significant for the nanocrystals, as detailed above. The B4e c/a experimental ratio is among the lowest known in the wurtzite phase, which supports both a high ionicity43 and, indirectly, its closeness to the still lower c/a value in the Bk phase, in which the two symmetric interlayer bonds shrink c and expand a. The equation of state, the V (p) relation, is commonly represented as the V /V0 versus p curve in the solid-state literature, with V0 being the V (p ) 0) value. Figure 7 displays this relation for the B4- and B1-like nanocrystals. In all cases, the nanocrystals of smaller sizes are less compressible than the
J. Phys. Chem. C, Vol. 112, No. 17, 2008 6673
Figure 7. Evolution of the relative volume V/V0 vs pressure (in GPa) for B1c, B4o, and B4of nanocrystals of selected sizes, together with the bulk behavior of the B1, Bk, and B4 phases (solid lines). V0 is the corresponding zero-pressure equilibrium volume.
Figure 8. B0 (in GPa) vs the inverse of the cubic root of the number of AlN molecules (1/N 1/3) in different nanocrystals. Symbols as in Figure 2. The solid B4n range indicates the value for the experimental AlN nanocrystals,8 while arrows mark the computed bulk limits for the B1, Bk, and B4 phases.
larger ones, although the spread is very small. The curves converge to a common one for large nanocrystals, which compare favorably as follows: B1c to B1b, B4o to Bkb, B4of to B4b. Again, the B4o nanocrystals have the Bk phase as their macroscopic limit. The B1 phase nanocrystals are the least compressible. On the other hand, fixing the internal coordinates makes the B4of nanocrystals more compressible than the more compact B4o ones. The V/V0-p relation is in turn usually described by an analytic form fitted to the experimental or theoretical values. A very successful form is that proposed by Vinet et al.;55 the adjustable parameters in this equation of state are the bulk modulus at zero pressure, B0 (B ) -V ∂p/∂V, inverse of the compressibility), and its first pressure derivative, B′0, also at zero pressure. The former is important from the technological point of view: the hardness of a given material usually correlates positively with its B value (negatively with its compressibility). The lowestcompressibility materials, diamond (443 GPa) and Os (462 GPa),56 need cheaper or special purpose replacements for some applications. The use of nanostructured materials is promising in this respect, since the compressibility may be quite different from that of the bulk materials. In this respect, AlN can be important, especially knowing the high hardness of the Bk phase of the isovalent BN material. Figure 8 shows the evolution of the B0 fitting parameter with the nanocrystal size, again by using the 1/N1/3 variable to ease the bulk extrapolation. Notice how the B1c and B1t curves, with very different behaviors, converge to the same bulk limit, with a very high B0 ) 363 GPa (B′0 ) 4.25). The B4o and B4h nanocrystals show similar behaviors, with values close to the experimental nanocrystals (B0 ) 321 GPa, ref 8), and share the same bulk limit, again the Bk phase with B0 ) 298 GPa (B′0 ) 4.21). On the other hand, the B4of and B4hf simulations have
6674 J. Phys. Chem. C, Vol. 112, No. 17, 2008 much lower bulk moduli, with a bulk limit even below the computed B4 one, B0 ) 272 GPa, but higher than the experimentally reported bulk,43 B0 ) 208 GPa. The behavior of the B4 solid seems to be quite odd, since our computed value for B′0 is 2.12 (unusually low), while the experimentally reported one, 6.3, is unusually high. A more recent study46 gives a B0 value of 204.4, although it does not give the B′0 value. Oddly enough, this same study does report both B0 and B′0 values for the high-pressure phase, 304 GPa and 3.9, respectively. Other theoretical and experimental (B0/GPa, B′0) pairs for the B1 phase are (295, 3.5),45 (221, 4.8),44 and (329, 5.0).57 It should be remarked, though, that estimating zero-pressure values of phases by extrapolating from high-pressure behavior is quite risky from the experimental point of view, thus explaining the rather widespread values. Overall, it seems plausible that the bulk value ordering is as given in Figure 8 and that our candidate nanocrystal series, B4o and B1c, converge from above to the B1 and Bk limits, respectively. Having this Bk phase, much less compressible than the B4 one, as limit, may also explain the behavior of the high experimentally measured B0 value for the B4-like nanocrystals, adding more support to our nonbuckled prediction for the nanocrystals. C. The B4-B1 Phase Transition in Nanocrystals. Experimentally, AlN presents a phase transition from the B4 phase to the B1 phase upon increasing pressure. However, and ignoring for the moment the different temperatures, different experiments display different transition pressures (ptr /GPa), 19.22,46 14,44 10.7-20,45 and 17.6-22.9.43 Despite some claims, all of these are direct transition pressures: they are obtained by measurement of the loading experiment, increasing pressure, which may differ from the unloading one. As presented in subsection II C, the equilibrium phase transition pressure is obtained by equating the Gibbs functions per unit formula of the two phases (both for the bulk and for a given nanocrystal size). However, experiments are usually far from quasistatic, thus displaying kinetic effects; there is a chemical potential barrier for the transition, and usually the transition will not start until a pressure larger than the equilibrium one (in the loading experiment) or lower than the equilibrium one (in the unloading experiment) is achieved. Sometimes, the high-pressure phase can even remain metastable at zero pressure.15,42 These phenomena are nothing but part of a hysteresis cycle. Experimentally, Xia et al.44 have retained the B1 phase of AlN back to room conditions; however, no other experimental work mentions the unloading process, and these authors’ B0 value for the B1 phase is unusually small. Our computed equilibrium transition pressure for the B4b h B1b bulk transition is ptr ) 4.56 GPa; given the lowest pressure at which the B1 phase appears (10.7 GPa, ref 45) as an upper ptr bound and the metastability at 0 GPa of the B1 phase (ref 44) as a lower ptr bound, our value seems to be reasonable. On the other hand, our corresponding prediction for the Bkb h B1b phase transition is higher, ptr ) 9.57 GPa. Given the fact that we predict a less stable B4b phase than it should, it seems plausible that our 4.56 GPa value for the B4b h B1b bulk transition is somewhat lower than the real one. Moving now to our nanocrystal values, the B4o h B1c phase transition pressure as a function of the nanocrystal size is depicted in Figure 9. First of all, notice that for small sizes ptr is negative: small B1c nanoclusters are lower in energy than B4o ones, and thus, the equilibrium condition has a negative pressure solution. This is easy to explain, since B1 atoms in the surface maintain a 5-fold coordination, while those in the B4- or Bk-like phases get at most 3- or 4-fold coordination,
Costales et al.
Figure 9. Evolution of the B4-like h B1-like phase transition equilibrium pressure (ptr, in GPa) with the nanocrystal size, as measured by N (number of AlN units, main axis) and by 1/N1/3 (inset). The inset also displays our computed values for several transitions among the bulk phases, the B4n experimental direct transition measured value for the nanocrystals (14.5, ref 8), and the r2 ) 0.99997 linear best-fit to our nanocrystal values, ptr ) 10.18 - 130.9/N1/3.
respectively, and hence, the former should be lower in energy when surface effects dominate. However, for sizes larger than N = 2000, the transition pressure becomes positive, displaying a continuously increasing trend. Although the final convergence cannot be seen in the main axes, using again the 1/N1/3 variable as an extrapolating means proves to be very fruitful. Indeed, the inset shows that the trend is almost perfectly linear and that the extrapolation to the bulk limit coincides very well with the Bkb h B1b phase transition pressure, as expected. It should be remarked that the above kind of behavior can be predicted by using the small-system model energy in eq 9 as follows. ptr(N) is determined by the equilibrium condition ∆G h (N, ptr) ) 0, which leads to
∆µ∞(ptr) ) -∆a(ptr)/N 1/3
(10)
from the large-N lines in Figure 5, we can infer that a(p), analogous to γ(p), varies in a similar way for the two phases and thus ∆a = constant. On the other hand, the bulk-limit chemical potentials are equal at p∞tr , the bulk-limit transition pressure, and thus to first order
h ∞tr (ptr - p∞tr ) ∆µ∞(ptr) ) ∆V
(11)
since V h ) ∂µ/∂p. Thus, to a good approximation,40
ptr ) p∞tr - N -1/3
∆a ∆V h ∞tr
(12)
which is the law observed here. Notice that both ∆a and ∆V h ∞tr are negative here, since the B1 phase is both the most compact (smaller V h ) and the one with lower surface energy per unit area (smaller γ and a), and hence, we infer that ∂ptr/∂N should be positive. Also, it is another thermodynamic result for small systems that ∂ptr/∂N ) -∆µtr/∆V h tr, where µ(N, p) ) (∂G/∂N)p, within the static limit. Since ∆V h tr < 0 in all cases and we see that the slope of ptr(N) is positive, µB1c(N, ptr) > µB4o(N, ptr). h , ptr f p∞tr , and thus both Note that, in the N f ∞ limit, µ∞ ) G chemical potentials must coincide: the slope of ptr(N) should go to zero, so that there is no size dependence in the macroscopic limit. Regarding the comparison with experimental results, the measured direct transition pressure for AlN nanocrystals is 14.5 GPa;8 it lies above our equilibrium ptr for the whole size range. Although it must also be above the thermodynamic equilibrium value due to hysteresis, one should notice that the hysteresis range should be smaller for nanocrystals than for bulk solids:
Buckled, Nonbuckled A1N Nanocrystal Structures it is likely that the transition proceeds on a single domain on each nanocrystal, thus avoiding domain boundary issues. This nanocrystal experimental ptr also lies above the lowest direct transition measured in the bulk, 10.7 GPa,45 but it is more or less in the same range as the rest. Recall that these direct transitions are upper bounds for the experimental equilibrium transition. Since the onset of a direct transition depends critically on sample preparation, pressure gradient with time, and other details of the experimental conditions, which differ in the various cited experiments, an assessment of the relative value of the thermodynamic equilibrium transition pressure of bulk and nanocrystal AlN is quite difficult. Regarding our computed equilibrium values, the increase of ptr with N is unequivocal, and it is most likely correct for the smallest sizes. The fact that its bulk limit is that of the Bkb h B1b transition may introduce another possibility: since we are certain that the bulk limit should be the B4b h B1b transition, there must be a threshold value for N from which B4-like structures (buckled, as opposed to the nonbuckled Bk-like ones that we have found thus far) dominate. The slope can change, even in sign, after this B4-like domination takes place. Since our fixed-geometry B4of and B4hf nanocrystals display many artifacts due to their nonequilibrium nature, we cannot make a good guess at the sign of the ptr(N) slope, lacking appropriate values to represent the B4-like phase. IV. Conclusions and Further Developments We have presented a recently developed theoretical scheme, the periodic cluster model, for the simulation of nanocrystals. The approach makes use of the limited periodicity displayed by these nanostructures, reducing the dimensionality of the problem to that of the periodically repeated unit cell, much in the spirit of periodic crystal calculations, but retaining the energetic surface effects. Its main drawback is that, requiring the cluster to be periodic, it neglects geometric surface relaxation or other nonperiodic effects, such as the spontaneous creation of domains, although combinations of periodic clusters may be devised to solve this problem. Due to its nature, including a periodic cell, it is easy to define a volume per formula unit; in this way, pressure effects can be incorporated into the scheme reliably, an extra benefit that compensates its drawbacks. This periodic cluster model has been implemented into an atomistic pair potential simulation code, together with a model we have developed for interactions on AlN nanoclusters. Through these, we have simulated AlN nanocrystals on a variety of structures, pressures, and sizes. Our results predict a nonbuckled structure, similar to the Bk graphitic-like structure displayed by, for example, bulk BN, instead of the buckled wurtzite-like structure (B4) displayed by bulk AlN. It is argued that the polar, buckled structure will eventually overcome the nonbuckled structure for a given threshold size, thus introducing a change in slope on the size behavior of the different properties (e.g., ptr(N)). Owing to its simplicity, the periodic cluster results abide quite well with the model energy expressions corresponding to small systems’ thermodynamics, leading to linear relations against 1/N1/3. We have computed surface tension, c/a ratio, and equation of state values against variations in both size and pressure. The obtained behavior is qualitatively correct, although the nonbuckled structure leads to Bk infinite-size limits instead of B4 ones. This is particularly important for the bulk modulus, a property which trends are usually related to the material’s hardness: the bulk Bk phase has a larger bulk modulus than the B4 one, and this leads also to nonbuckled (Bk-like)
J. Phys. Chem. C, Vol. 112, No. 17, 2008 6675 nanocrystals with a bulk modulus larger than that of the buckled (B4) bulk. As expected, the B1-like nanocrystals have an even larger bulk modulus. Regarding the phase transition, the spread of the bulk experimental values precludes a definite assignment to the sign of the experimental transition pressure variation with size. Our calculations unequivocally display an increase of the transition pressure with size, linear in 1/N1/3. This increase is justified through a small systems’ thermodynamic model, in which it is opposite to the ratio of two transition properties: the change in surface energies and the change in volumes per formula unit, which are both distinctly negative here. In addition to the intrinsic value of the results presented, this first application of the periodic cluster model suggests several areas for future work. On the one hand, our somewhat crude pair potential model may be substituted for more accurate ways of evaluating the potential energy surface. Although the computational cost will increase and the range of sizes available will decrease, other interesting information such as the electronic behavior may be obtained. On the other hand, neglecting the geometric surface relaxation can be a very important limitation in some cases. However, a generalized periodic cluster scheme can be sketched, in which the strictly 3D periodic nanocrystal is surrounded by 2D periodic slabs representing the surfaces, these in turn by 1D periodic rods representing edges, and explicit clusters representing vertices; its dimensionality will be somewhat larger but still on the same order of magnitude as the 3D periodic nanocrystal. Yet another generalization is currently under way in our laboratory, in which we employ the same periodic approach to the treatment of other kinds of nanostructures, such as nanobelts and nanorings. Finally, our finding of nonbuckled structures for AlN nanocrystals seems to be genuine for those of small size. Although we cannot give an actual number for the buckling threshold size, this behavior can be experimentally tested: since the Bk bulk structure does display a symmetry center and is nonpolar, nonbuckled nanocrystals should have characteristic diffraction patterns quite different from the polar, non-centrosymmetric, buckled ones. We believe that this competition will also be present in other nanostructures in AlN and similar compounds, such as the recently reported ZnO nanobelts and nanorings.47 Acknowledgment. This research has been funded by the Spanish Ministerio de Educacio´n y Ciencia (MEC), Grants BQU2003-06553 and CTQ2006-02976, cofinanced by the European Regional Development Fund (FEDER). A.C. wishes to thank the Spanish MEC and the Fondo Social Europeo for her Ramo´n y Cajal position. C.J.F.S. thanks the Spanish MEC for his F. P. I. grant. We want to thank Prof. John Vail for his kind help with the shell model calculations. We dedicate this work to the memory of our late Professor, colleague, and friend, Prof. Lorenzo Pueyo. References and Notes (1) Alivisatos, A. P. J. Phys. Chem. 1996, 100, 13226-13239. (2) Goldstein, A. N.; Echer, C. M.; Alivisatos, A. P. Science 1992, 256, 1425-1427. (3) Qadri, S. B.; Yang, J.; Ratna, B. R.; Skelton, E. F.; Hu, J. Z. Appl. Phys. Lett. 1996, 69, 2205-2207. (4) Chen, C. C.; Herhold, A. B.; Johnson, C. S.; Alivisatos, A. P. Science 1997, 276, 398-401. (5) Jiang, J. Z.; Staun Olsen, J.; Gerward, L.; Morup, S. Europhys. Lett. 1998, 44, 620-626. (6) Qadri, S. B.; Skelton, E. F.; Dinsmore, A. D.; Hu, J. Z.; Kim, W. J.; Nelson, C.; Ratna, B. R. J. Appl. Phys. 2001, 89, 115-119. (7) Jacobs, K.; Zaziski, D.; Scher, E. C.; Herhold, A. B.; Alivisatos, A. P. Science 2001, 293, 1803-1806.
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